WASHINGTON — Speaker John Boehner failed to harness his unruly House majority Thursday night and canceled a planned vote on a bill to ­extend income tax cuts for most Americans while allowing rates to rise on millionaires.

The turn of events appeared to weaken Boehner’s hand in budget ­negotiations with the White House. If his bold gambit had passed, the Republican leader would have gained a fresh bargaining chip in talks with President Obama. But the staunch resistance of conservative Tea Party representatives to tax increases of any kind — despite closed-door entreaties by the speaker — foiled that plan, and his move, dubbed “Plan B,” backfired.

Now the most significant chance for America to avoid a potentially devastating $500 billion in automatic tax rate increases and spending cuts on Jan. 1 — the so-called fiscal cliff — is for Obama and Boehner to return to the negotiating table and hammer out a deal that can attract the support of a coalition of House Democrats and more moderate Republicans.

It was not immediately clear Thursday night how or when negotiations toward such a solution might continue. Boehner told his colleagues in a closed meeting that he would resume negotiations with Obama, the Associated Press reported.

There was no confirmation from the White House, which issued only a brief statement after the lack of action in the House: “We are hopeful that we will be able to find a bipartisan solution quickly that protects the middle class and our economy.’’

Congress planned to depart for a holiday break Thursday night after the House leadership suddenly abandoned its effort after a day of cajoling and vote-wrangling fell short. The Capitol is scheduled to go quiet until the day after Christmas.

“Between now and next Wednesday, I am hopeful that the president, the speaker, and others will work together to get an agreement that can be adopted by the House of Representatives,’’ Steny Hoyer, the House minority whip, said on MSNBC Thursday night. “That would require what we have not seen from the Republican majority — a willingness to compromise, and be reasonable, and make sure the math works.’’

‘I did my part, they’ve done nothing.’

Hoyer called the speaker’s stumble, virtually a week before the fiscal cliff, a “debacle.’’ Boehner issued a terse statement after he failed to round up enough votes from his representatives.

“The House did not take up the tax measure today because it did not have sufficient support from our members to pass,’’ he said.

The speaker said it was now up to the Senate to act to avert the fiscal cliff, apparently ­acknowledging that the House GOP leadership is unable to develop any further solution on its own. He called on the Senate to approve a House-passed measure to extend income tax cuts for all taxpayers — including the wealthy. That measure has been staunchly opposed by Obama and the Senate leadership.

Boehner’s decision to pursue “Plan B’’ had triggered a fresh round of heated exchanges and brought negotiations to a standstill this week after progress last weekend and on Monday.

Boehner’s measure would have preserved Bush-era tax cuts for the vast majority of Americans while allowing them to expire for people earning more than $1 million. As a practical matter, the bill was doomed. Senate Democrats said they would not approve it, and President Obama pledged a veto if it reached his desk.

The failure demonstrates the extent to which conservative Republicans are entrenched against any tax hike at all. Republicans have spent years running election campaigns against tax increases, and Boehner was seeking a big concession from his members as he sought to win a breakthrough on the fiscal cliff. ­Under his “Plan B,’’ he said, income taxes rates would remain the same for 99.81 percent of taxpayers, compared with 98 percent under Obama’s earlier proposal.

Boehner expressed frustration earlier in the day that he had not won more concessions on spending from the White House in earlier negotiations, after he said over the weekend that he would be willing to ­allow taxes to rise on millionaires.

“For weeks, the White House said if I moved on rates, that they would make substantial concessions on spending cuts and entitlement reform,” Boehner said in an afternoon news conference. “I did my part, they’ve done nothing.”

Failing on the planned Thursday vote also strips away potential political cover for ­Republicans, who polls show will get the lion’s share of the blame in the event negotiations cannot rescue the nation from the fiscal cliff. Public anger increased toward Republicans and the Tea Party in 2011 when conservatives refused to raise the nation’s borrowing limit, and the party is risking a similar backlash.

The White House earlier Thursday criticized Boehner’s unilateral effort as a setback to negotiations.

‘‘Instead of taking the opportunity that was presented to them, to continue to negotiate what could be a very helpful large deal for the American people, the Republicans in the House have decided to run down an alley that has no exit while we all watch,’’
said White House press secretary Jay Carney. ‘‘And again it’s something we’ve seen in the past.’’

Obama’s latest position in the talks has been to allow taxes to rise on families earning more than $400,000 a year, a concession from his earlier position that people making more than $250,000 a year should pay more. Obama has also signalled a willingness to reduce the growth of Social Security benefits, but changing the cost-of-living index that is used to calculate annual increases.

Earlier in the week, in terms of overall proposals, the two sides also had narrowed their financial differences. The president had scaled back his proposal for tax revenues to $1.3 trillion over a decades, coupled with $930 billion in spending cuts. Boehner had sought a dollar-for-dollar pairing of cuts and revenues, set at $1 trillion.

Some Republicans who backed Boehner said they hoped the urgent need for a deal would persuade all parties to back the tax increase for millionaires, including Democrats.

“We have to get something over to the Senate to strengthen the Speaker’s position,” said Representative Charlie Dent, a Pennsylvania Republican.

But Democrats branded Boehner’s bill a political ­maneuver that will not fool voters.

“The ‘B’ in the Republican so-called Plan B stands for ­bogus, boondoggle, and bluff,’’ said US Representative Edward Markey, Democrat of Malden. The political forces weighing on Boehner are considerable. He has a number of Tea Party Republicans in his caucus who are staunchly opposed to compromise.

To avoid a leadership fight that could result in his removal as speaker, he needed to find a way to balance the demands of those members and the realities of the fiscal cliff. Now his ­caucus has left him with few options.

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